Q. 19 - Intermediate Algebra

Question

In the following exercises, estimate each root between two consecutive whole numbers. 

ⓐ $\sqrt{70}$ ⓑ $\sqrt[3]{71}$ 

Step-by-Step Solution

Answer

Part (a) $\Rightarrow \sqrt{70}=8<\sqrt{70}<9$

Part (b)$\Rightarrow \sqrt[3]{71}=4<\sqrt[3]{71}<5$

Part (a) Step 1. Given information

(a)$\sqrt{70}$

Estimate each root between two consecutive whole numbers.

Step 2. Explanation

Think of the perfect square numbers closest to$70$.

 Make a small table of these perfect squares and their squares roots.
$\begin{array}{ccc}\text{Number}& \text{Square root}& \\ 64& 8& \\ \begin{array}{l}81\\ 100\end{array}& \begin{array}{l}9\\ 10\end{array}& \end{array}$

Locate$70$ between two consecutive perfect squares.

$64<70<81$

$\sqrt{70}$ is between their square roots. $8<\sqrt{70}<9$

Part (b) Step 3 .Given information

(b) $\sqrt[3]{71}$

Estimate each root between two consecutive whole numbers.

Step 4. Explanation

Similarly, we locate $71$between two perfect cube numbers.

$\begin{array}{cc}\text{Number}& \text{Cube root}\\ \begin{array}{l}64\\ 125\end{array}& \begin{array}{l}4\\ 5\end{array}\end{array}$

Locate $71$between two consecutive perfect cubes.

$\Rightarrow 64<71<125$

$\sqrt[3]{71}$ is between their cube roots.

$4<\sqrt[3]{71}<5$